3 SPECTRUM OF A COMMUTATIVE BANACH ALGEBRA 323 Z is not empty: if Z were empty there would exist a finite open covering (Vj) of X, and for each j a function xj e m such that */(/) =£ 0 for all t e Vj; now consider the function ]£ Xj Xj , which belongs to m.) (d) Deduce from (b) and (c) that the spectrum X(A) of the algebra of Section 15.1, Problem 1 can be canonically identified with the interval [0, 1]. 2. Show that the spectrum of the algebra A of Section 15.1, Problem 2 consists of a single point, and that the unique maximal ideal of A is the radical of A (Section 15.2, Problem 7). 3. In the algebra sf(X) of Section 15.1, Problem 3, show that \\xn\\ ^ ||*||B/(JI - 1)!, and deduce that ^(X) is equal to its radical. Deduce that J/(X) has no characters (use Problem 5 of Section 15.1). 4. Let B be the Banach subalgebra of ^c^P) (notation of (15.3.8)) generated by and the function |/0| : £h-> |£|. Show that X(B) is homeomorphic to the set of points (*i, x2 , *3) e R3 such that x\ -f xl ^ x§ and 0 ^ *3 ^ 1. (Remark that B contains all functions of the form £i— »0(1£|), where g is a continuous function on [0, 1], and that B also contains the subalgebra A0 of .^(D) introduced in (15.3.8). To show that lx(/o)l ^ X(l/ol) for all characters x of B» consider the function £(£ - (|£| + e))"1, which belongs to B for all e > 0. 5. The space /£ (6.5) becomes a Banach algebra without unit element under the multipli- cation (^nX7?*) — (£nr)n)> Let A be the Banach algebra obtained by adjoining a unit element to /£ (Section 15.1, Problem 5). Show that X(A) may be identified with the compact subset of R consisting of +00 and the integers ^1 (use the fact that /£ can be canonically identified with its dual). The Gelfand transformation ^A then becomes the identity mapping, and ^A(A) is a nonclosed dense subalgebra of (a) Show that the dual of the underlying Banach space of the Beurling algebra A (Section 15.1, Problem 4) can be identified with the space of classes of A-measurable complex-valued functions g such that \\9\\2 = sup where the supremum is taken over the set of functions w e Ci such that V(to) > 0. Then \\g\\ is the norm on the dual A' of A. Show that n Zn -T- l The canonical bilinear form </, #> (where /e A and g e A') is then equal to (b) Deduce from (a) that the characters of the Banach algebra A obtained by ad- joining a unit element to A (Section 15.1, Problem 5) have as their restrictions to A) ^ 0 for all t e X, then the inverse x"1 of x in ^C(X) belongs to